This lesson is designed to help students learn about algorithms through
Venn Diagrams and
Box Plots. Students will develop algorithms for solving Venn Diagrams, collect data for each algorithm, and
compare the efficiency of each algorithm, using box plots.

Objectives

Upon completion of this lesson, students will:

understand the purpose and use of algorithms in problem-solving

understand what algorithms are most efficient for solving Venn Diagrams

understand an authentic application of box plots in data analysis

Standards Addressed:

Grade 10

Process Skills

The student demonstrates an ability to problem solve.

The student communicates his or her mathematical thinking.

The student demonstrates an ability to use logic and reason.

Grade 3

Process Skills

The student demonstrates an ability to problem solve.

The student demonstrates an ability to use logic and reason.

Grade 4

Process Skills

The student demonstrates an ability to problem solve.

The student demonstrates an ability to use logic and reason.

Grade 5

Functions and Relationships

The student communicates his or her mathematical thinking.

The student demonstrates an ability to use logic and reason.

Process Skills

The student demonstrates an ability to problem solve.

The student demonstrates an ability to use logic and reason.

Grade 6

Process Skills

The student demonstrates an ability to problem solve.

The student communicates his or her mathematical thinking.

The student demonstrates an ability to use logic and reason.

Grade 7

Process Skills

The student demonstrates an ability to problem solve.

The student communicates his or her mathematical thinking.

The student demonstrates an ability to use logic and reason.

Grade 8

Process Skills

The student demonstrates an ability to problem solve.

The student communicates his or her mathematical thinking.

The student demonstrates an ability to use logic and reason.

Grade 9

Process Skills

The student demonstrates an ability to problem solve.

The student communicates his or her mathematical thinking.

The student demonstrates an ability to use logic and reason.

Grade 6

Mathematical Reasoning

1.0 Students make decisions about how to approach problems

2.0 Students use strategies, skills, and concepts in finding solutions

Statistics, Data Analysis, and Probability

1.0 Students compute and analyze statistical measurements for data sets

Grade 7

Mathematical Reasoning

1.0 Students make decisions about how to approach problems

2.0 Students use strategies, skills, and concepts in finding solutions

Grades 8-12

AP Probability and Statistics

14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

Probability and Statistics

8.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

Statistics and Probability

Interpreting Categorical and Quantitative Data

Summarize, represent, and interpret data on a single count or measurement variable

Grades 6-8

Data Analysis and Probability

Develop and evaluate inferences and predictions that are based on data

Grades 9-12

Data Analysis and Probability

Develop and evaluate inferences and predictions that are based on data

Grade 7

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 4: The learner will understand and use graphs and data analysis.

Grade 8

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 4: The learner will understand and use graphs and data analysis.

Introductory Mathematics

Data Analysis and Probability

COMPETENCY GOAL 3: The learner will understand and use graphs and data analysis.

Technical Mathematics I

Number and Operations

Competency Goal 1: The learner will apply various strategies to solve problems.

6th Grade

Mathematical Processes

The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

7th Grade

Mathematical Processes

The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

8th grade

Mathematical Processes

The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.

6th Grade

Data Analysis & Probability

Content Standard 5.0 The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

7th Grade

Data Analysis & Probability

The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

8th Grade

Data Analysis & Probability

The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

Grade 6

Probability and Statistics

10. The student uses statistical representations to analyze data.

Underlying Processes and Mathematical Tools

12. The student communicates about Grade 6
mathematics through informal and mathematical language, representations, and models.

Grade 7

Underlying Processes and Mathematical Tools

14. The student communicates about Grade 7
mathematics through informal and mathematical language, representations, and models.

Grade 8

Probability and Statistics

12. The student uses statistical procedures to describe data.

Underlying Processes and Mathematical Tools

15. The student communicates about Grade 8
mathematics through informal and mathematical language, representations, and models.

6th Grade

Probability and Statistics

6.18c The student, given a problem situation, will collect, analyze, display, and interpret data in a variety of graphical methods, including box-and-whisker plots.

7th Grade

Probability and Statistics

7.17 The student, given a problem situation, will collect, analyze, display, and interpret data, using a variety of graphical methods, including frequency distributions; line plots; histograms; stem-and-leaf plots; box-and-whisker plots; and scattergrams.

8th Grade

Probability and Statistics

8.12 The student will make comparisons, predictions, and inferences, using information displayed in frequency distributions; box-and-whisker plots; scattergrams; line, bar, circle, and picture graphs; and histograms.

Reason for Alignment: This lesson motivates understanding of algorithm through the use of Venn diagrams. The use of Venn diagrams and a rule for these diagrams goes with the algorithm part of the section.

A copy of the
worksheet for each student (optional)

pencil and paper

Key Terms

algorithm

Step-by-step procedure by which an operation can be carried out

box plot

Also called box-and-whisker plot, this graph shows the distribution of data by dividing the data into four groups with the same number of data points in each group. The box contains the middle 50% of the data points and each of the two whiskers contain 25% of the data points.

Venn Diagram

A diagram where sets are represented as simple geometric figures, with overlapping and similarity of sets represented by intersections and unions of the figures

Lesson Outline

Focus and Review

Open the
Venn Diagram Shape Sorter applet, and project it for the whole class to view. Set the applet to
Guess Mode, and work through a problem as a class:

Guide the students through the activity by instructing them on applet functionality.

Have students volunteer answers and tell you where to place different shapes in order to solve
the problem.

Ask guided questions to help students start thinking about why they're selecting shapes to
try.

After solving the problem, ask students to explain what approach they took in solving it.

Ask guided questions to help students start thinking about efficiency in solving problems

Objectives

Lead the class in a
discussion on algorithms. Explain to the class that they will be finding algorithms to solve Venn Diagram problems, and
collecting data and using box plots in order to determine an algorithm's efficiency.

Introduce the
Box Plot activity, showing students how to create box plots for different categories of data.

Guided Practice

Explain how you'll use the box plot to compare different algorithms:

Students will develop an algorithm for solving a Venn Diagram

Students will solve several different problems using that method, recording how many shapes
they had to try before being able to accurately guess the rules.

The class will come together and enter their data into a box plot, with each different
algorithm as a category.

If necessary, practice this process as a class until you're confident that students can do it on
their own.

Remind students to develop an algorithm for solving the problems.

Have students record how many shapes they try before being able to solve the problem each
time.

If students are having trouble with the concept of algorithms, have them use the
worksheet to guide their thinking.

Closure

As a class, enter data into the
box plot, where each different algorithm is a different category in the box plot. Discuss the results:

Which algorithm is most efficient?

How can you tell?

How does the box plot help you see this?

Alternate Outline

If only one computer is available for the classroom, this lesson can be rearranged in the
following way:

The teacher may do this activity as a demonstration. Choose the version (one circle, etc.) and
allow students to decide individually, or in groups, which object to move onto the diagram and
where to move it.

As a class, construct a box plot of the results of each algorithm. Then discuss conclusions
they can draw from their results.

Suggested Follow-Up

If students need more work with displaying and analyzing data, you may want to use the lessons
Mean, Median and Mode or
Box Plots.